Optimization of shape functionals under convexity, diameter or constant width constraints is challenging from a numerical point of view. The support and gauge functions allow a functional characterization of these constraints. Functions describing convex sets are discretized using truncated spectral decompositions or values on a uniform grid. I will present the resulting numerical frameworks, together with various applications from convex geometry and spectral optimization.
In the second part of the talk, I will discuss a different point of view on constrained optimization problems. The Blaschke-Santalo diagrams can completely characterize all possible inequalities between various quantites, under eventual constraints. The complete theoretical characterization of such diagrams is often difficult to obtain, motivating the interest in algorithms allowing their numerical approximations. Recent developments related to this topic, with applications in algebra and convex geometry, will be presented. In this process, yet another discretization process for convex shapes is proposed.